提出新方法，精准删除数据同时保护其他知识不被误伤

Machine unlearning has reached a critical bottleneck. As traditional weight-space interventions focus primarily on erasing targeted concepts, they often fail to prevent the unintended suppression of other significant representations. This leads to substantial collateral damage, with essential knowledge being forgotten, because these methods lack formal mathematical guarantees for the preservation of neutral concepts. To avoid degradation, they are frequently forced into conservative updates. We propose BARRIER (Bounded Activation Regions for Robust Information Erasure), a paradigm-shifting framework that shifts the locus of intervention from static model weights to the dynamic geometry of hidden-layer activations. Unlike existing methods, BARRIER employs Interval Arithmetic (IA) on SVD-based projections of the activation space to encapsulate the specific target region within a bounding hypercube. By driving unlearning updates exclusively within this forget interval and mathematically bounding the model response on the complement, we ensure rigorous protection of the retain distribution. This geometric construction transforms the preservation of knowledge from an empirical heuristic into a formal optimization target with a probabilistic tail bound on functional drift. Crucially, this stability permits highly aggressive unlearning updates within the forget region. Empirical evaluations demonstrate that BARRIER matches state-of-the-art trade-offs across classifiers and diffusion models, maximizing targeted concept erasure while safeguarding the integrity of all other representations. Our code is available at https://github.com/OneAndZero24/BARRIER.